chapter after this one I will describe generalizations of customary objects of general topology
described in this chapter.

The reason why I've written this chapter is to show to the reader kinds of objects which I

generalize below in this book. For example, funcoids and a generalization of proximity spaces, and
funcoids are a generalization of pretopologies. To understand the intuitive meaning of funcoids one
needs rst know what are proximities and what are pretopologies.

Having said that, customary topology is

not

used in my denitions and proofs below. It is just

to feed your intuition.

5.1 Metric spaces

The theory of topological spaces started immediately with the denition would be completely
non-intuitive for the reader. It is the reason why I rst describe metric spaces and show that
metric spaces give rise for a topology (see below). Topological spaces are understandable as a
generalization of topologies induced by metric spaces.